Implementing quantum gates using the ferromagnetic spin-J XXZ chain with kink boundary conditions
Identifieur interne : 001373 ( Istex/Corpus ); précédent : 001372; suivant : 001374Implementing quantum gates using the ferromagnetic spin-J XXZ chain with kink boundary conditions
Auteurs : Tom Michoel ; Jaideep Mulherkar ; Bruno NachtergaeleSource :
- New Journal of Physics [ 1367-2630 ] ; 2010.
Abstract
We demonstrate an implementation scheme for constructing quantum gates using unitary evolutions of the one-dimensional spin-J ferromagnetic XXZ chain. We present numerical results based on simulations of the chain using the time-dependent density matrix renormalization group method and techniques from optimal control theory. Using only a few control parameters, we find that it is possible to implement one- and two-qubit gates on a system of spin- XXZ chains, such as Not, Hadamard, Pi-8, Phase and C-Not, with fidelity levels exceeding 99.
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DOI: 10.1088/1367-2630/12/2/025003
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Implementing quantum gates using the ferromagnetic spin-J XXZ chain with kink boundary conditions</title><author><name sortKey="Michoel, Tom" sort="Michoel, Tom" uniqKey="Michoel T" first="Tom" last="Michoel">Tom Michoel</name><affiliation><mods:affiliation>Department of Plant Systems Biology, VIB, Department of Molecular Genetics, Ghent University, Technologiepark 927, B-9052 Gent, Belgium</mods:affiliation></affiliation></author><author><name sortKey="Mulherkar, Jaideep" sort="Mulherkar, Jaideep" uniqKey="Mulherkar J" first="Jaideep" last="Mulherkar">Jaideep Mulherkar</name><affiliation><mods:affiliation>Department of Mathematics, University of California, Davis, CA 95616-8633, USA</mods:affiliation></affiliation></author><author><name sortKey="Nachtergaele, Bruno" sort="Nachtergaele, Bruno" uniqKey="Nachtergaele B" first="Bruno" last="Nachtergaele">Bruno Nachtergaele</name><affiliation><mods:affiliation>Department of Mathematics, University of California, Davis, CA 95616-8633, USA</mods:affiliation></affiliation><affiliation><mods:affiliation>Author to whom any correspondence should be addressed.</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: bxn@math.ucdavis.edu</mods:affiliation></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">ISTEX</idno><idno type="RBID">ISTEX:443D39C954A16B7BB8BD220B153760B4AC0C0A86</idno><date when="2010" year="2010">2010</date><idno type="doi">10.1088/1367-2630/12/2/025003</idno><idno type="url">https://api.istex.fr/document/443D39C954A16B7BB8BD220B153760B4AC0C0A86/fulltext/pdf</idno><idno type="wicri:Area/Istex/Corpus">001373</idno></publicationStmt><sourceDesc><biblStruct><analytic><title level="a" type="main" xml:lang="en">Implementing quantum gates using the ferromagnetic spin-J XXZ chain with kink boundary conditions</title><author><name sortKey="Michoel, Tom" sort="Michoel, Tom" uniqKey="Michoel T" first="Tom" last="Michoel">Tom Michoel</name><affiliation><mods:affiliation>Department of Plant Systems Biology, VIB, Department of Molecular Genetics, Ghent University, Technologiepark 927, B-9052 Gent, Belgium</mods:affiliation></affiliation></author><author><name sortKey="Mulherkar, Jaideep" sort="Mulherkar, Jaideep" uniqKey="Mulherkar J" first="Jaideep" last="Mulherkar">Jaideep Mulherkar</name><affiliation><mods:affiliation>Department of Mathematics, University of California, Davis, CA 95616-8633, USA</mods:affiliation></affiliation></author><author><name sortKey="Nachtergaele, Bruno" sort="Nachtergaele, Bruno" uniqKey="Nachtergaele B" first="Bruno" last="Nachtergaele">Bruno Nachtergaele</name><affiliation><mods:affiliation>Department of Mathematics, University of California, Davis, CA 95616-8633, USA</mods:affiliation></affiliation><affiliation><mods:affiliation>Author to whom any correspondence should be addressed.</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: bxn@math.ucdavis.edu</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">New Journal of Physics</title><title level="j" type="abbrev">New J. Phys.</title><idno type="ISSN">1367-2630</idno><idno type="eISSN">1367-2630</idno><imprint><publisher>IOP Publishing</publisher><date type="published" when="2010">2010</date><biblScope unit="volume">12</biblScope><biblScope unit="issue">2</biblScope><biblScope unit="page" from="1">1</biblScope><biblScope unit="page" to="15">15</biblScope></imprint><idno type="ISSN">1367-2630</idno></series><idno type="istex">443D39C954A16B7BB8BD220B153760B4AC0C0A86</idno><idno type="DOI">10.1088/1367-2630/12/2/025003</idno><idno type="articleID">317881</idno><idno type="articleNumber">025003</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">1367-2630</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract">We demonstrate an implementation scheme for constructing quantum gates using unitary evolutions of the one-dimensional spin-J ferromagnetic XXZ chain. We present numerical results based on simulations of the chain using the time-dependent density matrix renormalization group method and techniques from optimal control theory. Using only a few control parameters, we find that it is possible to implement one- and two-qubit gates on a system of spin- XXZ chains, such as Not, Hadamard, Pi-8, Phase and C-Not, with fidelity levels exceeding 99.</div></front></TEI><istex><corpusName>iop</corpusName><author><json:item><name>Tom Michoel</name><affiliations><json:string>Department of Plant Systems Biology, VIB, Department of Molecular Genetics, Ghent University, Technologiepark 927, B-9052 Gent, Belgium</json:string></affiliations></json:item><json:item><name>Jaideep Mulherkar</name><affiliations><json:string>Department of Mathematics, University of California, Davis, CA 95616-8633, USA</json:string></affiliations></json:item><json:item><name>Bruno Nachtergaele</name><affiliations><json:string>Department of Mathematics, University of California, Davis, CA 95616-8633, USA</json:string><json:string>Author to whom any correspondence should be addressed.</json:string><json:string>E-mail: bxn@math.ucdavis.edu</json:string></affiliations></json:item></author><language><json:string>eng</json:string></language><originalGenre><json:string>paper</json:string></originalGenre><abstract>We demonstrate an implementation scheme for constructing quantum gates using unitary evolutions of the one-dimensional spin-J ferromagnetic XXZ chain. We present numerical results based on simulations of the chain using the time-dependent density matrix renormalization group method and techniques from optimal control theory. Using only a few control parameters, we find that it is possible to implement one- and two-qubit gates on a system of spin- XXZ chains, such as Not, Hadamard, Pi-8, Phase and C-Not, with fidelity levels exceeding 99.</abstract><qualityIndicators><score>6.472</score><pdfVersion>1.4</pdfVersion><pdfPageSize>595.276 x 841.89 pts (A4)</pdfPageSize><refBibsNative>true</refBibsNative><keywordCount>0</keywordCount><abstractCharCount>543</abstractCharCount><pdfWordCount>5395</pdfWordCount><pdfCharCount>26860</pdfCharCount><pdfPageCount>15</pdfPageCount><abstractWordCount>81</abstractWordCount></qualityIndicators><title>Implementing quantum gates using the ferromagnetic spin-J XXZ chain with kink boundary conditions</title><genre><json:string>research-article</json:string></genre><host><volume>12</volume><publisherId><json:string>nj</json:string></publisherId><pages><total>15</total><last>15</last><first>1</first></pages><issn><json:string>1367-2630</json:string></issn><issue>2</issue><genre><json:string>journal</json:string></genre><language><json:string>unknown</json:string></language><eissn><json:string>1367-2630</json:string></eissn><title>New Journal of Physics</title></host><publicationDate>2010</publicationDate><copyrightDate>2010</copyrightDate><doi><json:string>10.1088/1367-2630/12/2/025003</json:string></doi><id>443D39C954A16B7BB8BD220B153760B4AC0C0A86</id><score>0.23089968</score><fulltext><json:item><original>true</original><mimetype>application/pdf</mimetype><extension>pdf</extension><uri>https://api.istex.fr/document/443D39C954A16B7BB8BD220B153760B4AC0C0A86/fulltext/pdf</uri></json:item><json:item><original>false</original><mimetype>application/zip</mimetype><extension>zip</extension><uri>https://api.istex.fr/document/443D39C954A16B7BB8BD220B153760B4AC0C0A86/fulltext/zip</uri></json:item><istex:fulltextTEI uri="https://api.istex.fr/document/443D39C954A16B7BB8BD220B153760B4AC0C0A86/fulltext/tei"><teiHeader><fileDesc><titleStmt><title level="a" type="main" xml:lang="en">Implementing quantum gates using the ferromagnetic spin-J XXZ chain with kink boundary conditions</title></titleStmt><publicationStmt><authority>ISTEX</authority><publisher>IOP Publishing</publisher><availability><p>IOP Publishing and Deutsche Physikalische Gesellschaft</p></availability><date>2010</date></publicationStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">Implementing quantum gates using the ferromagnetic spin-J XXZ chain with kink boundary conditions</title><author xml:id="author-1"><persName><forename type="first">Tom</forename><surname>Michoel</surname></persName><affiliation>Department of Plant Systems Biology, VIB, Department of Molecular Genetics, Ghent University, Technologiepark 927, B-9052 Gent, Belgium</affiliation></author><author xml:id="author-2"><persName><forename type="first">Jaideep</forename><surname>Mulherkar</surname></persName><affiliation>Department of Mathematics, University of California, Davis, CA 95616-8633, USA</affiliation></author><author xml:id="author-3"><persName><forename type="first">Bruno</forename><surname>Nachtergaele</surname></persName><email>bxn@math.ucdavis.edu</email><affiliation>Department of Mathematics, University of California, Davis, CA 95616-8633, USA</affiliation><affiliation>Author to whom any correspondence should be addressed.</affiliation></author></analytic><monogr><title level="j">New Journal of Physics</title><title level="j" type="abbrev">New J. Phys.</title><idno type="pISSN">1367-2630</idno><idno type="eISSN">1367-2630</idno><imprint><publisher>IOP Publishing</publisher><date type="published" when="2010"></date><biblScope unit="volume">12</biblScope><biblScope unit="issue">2</biblScope><biblScope unit="page" from="1">1</biblScope><biblScope unit="page" to="15">15</biblScope></imprint></monogr><idno type="istex">443D39C954A16B7BB8BD220B153760B4AC0C0A86</idno><idno type="DOI">10.1088/1367-2630/12/2/025003</idno><idno type="articleID">317881</idno><idno type="articleNumber">025003</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>2010</date></creation><langUsage><language ident="en">en</language></langUsage><abstract><p>We demonstrate an implementation scheme for constructing quantum gates using unitary evolutions of the one-dimensional spin-J ferromagnetic XXZ chain. We present numerical results based on simulations of the chain using the time-dependent density matrix renormalization group method and techniques from optimal control theory. Using only a few control parameters, we find that it is possible to implement one- and two-qubit gates on a system of spin- XXZ chains, such as Not, Hadamard, Pi-8, Phase and C-Not, with fidelity levels exceeding 99.</p></abstract></profileDesc><revisionDesc><change when="2010">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><original>false</original><mimetype>text/plain</mimetype><extension>txt</extension><uri>https://api.istex.fr/document/443D39C954A16B7BB8BD220B153760B4AC0C0A86/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="corpus iop not found" wicri:toSee="no header"><istex:xmlDeclaration>version="1.0" encoding="ISO-8859-1"</istex:xmlDeclaration><istex:docType SYSTEM="http://ej.iop.org/dtd/iopv1_5_2.dtd" name="istex:docType"></istex:docType><istex:document><article artid="nj317881"><article-metadata><jnl-data jnlid="nj"><jnl-fullname>New Journal of Physics</jnl-fullname><jnl-abbreviation>New J. Phys.</jnl-abbreviation><jnl-shortname>NJP</jnl-shortname><jnl-issn>1367-2630</jnl-issn><jnl-coden>NJOPFM</jnl-coden><jnl-imprint>IOP Publishing</jnl-imprint><jnl-web-address>stacks.iop.org/NJP</jnl-web-address></jnl-data><volume-data><year-publication>2010</year-publication><volume-number>12</volume-number></volume-data><issue-data><issue-number>2</issue-number><coverdate>February 2010</coverdate></issue-data><article-data><article-type type="paper" sort="regular"></article-type><type-number type="paper" numbering="article" artnum="025003">025003</type-number><article-number>317881</article-number><first-page>1</first-page><last-page>15</last-page><length>15</length><pii></pii><doi>10.1088/1367-2630/12/2/025003</doi><copyright>IOP Publishing and Deutsche Physikalische Gesellschaft</copyright><ccc>1367-2630/10/025003+15$30.00</ccc><printed></printed></article-data></article-metadata><header><title-group><title>Implementing quantum gates using the ferromagnetic spin-<inline-eqn><math-text><italic>J</italic></math-text></inline-eqn> XXZ chain with kink boundary conditions</title><short-title>Implementing quantum gates using the ferromagnetic spin-<inline-eqn><math-text><italic>J</italic></math-text></inline-eqn> XXZ chain</short-title><ej-title>Implementing quantum gates using the ferromagnetic spin-J XXZ chain with kink boundary conditions</ej-title></title-group><author-group><author address="nj317881ad1"><first-names>Tom</first-names><second-name>Michoel</second-name></author><author address="nj317881ad2"><first-names>Jaideep</first-names><second-name>Mulherkar</second-name></author><author address="nj317881ad2" alt-address="nj317881aad3" email="nj317881ea1"><first-names>Bruno</first-names><second-name>Nachtergaele</second-name></author><short-author-list>Tom Michoel <italic>et al</italic></short-author-list></author-group><address-group><address id="nj317881ad1" showid="yes">Department of Plant Systems Biology, VIB, Department of Molecular Genetics, <orgname>Ghent University</orgname>, Technologiepark 927, B-9052 Gent, <country>Belgium</country></address><address id="nj317881ad2" showid="yes">Department of Mathematics, <orgname>University of California</orgname>, Davis, CA 95616-8633, <country>USA</country></address><address id="nj317881aad3" alt="yes" showid="yes">Author to whom any correspondence should be addressed.</address><e-address id="nj317881ea1"><email mailto="bxn@math.ucdavis.edu">bxn@math.ucdavis.edu</email></e-address></address-group><history received="30 April 2009" online="26 February 2010"></history><abstract-group><abstract><heading>Abstract</heading><p indent="no">We demonstrate an implementation scheme for constructing quantum gates using unitary evolutions of the one-dimensional spin-<inline-eqn><math-text><italic>J</italic></math-text></inline-eqn> ferromagnetic XXZ chain. We present numerical results based on simulations of the chain using the time-dependent density matrix renormalization group method and techniques from optimal control theory. Using only a few control parameters, we find that it is possible to implement one- and two-qubit gates on a system of spin-<inline-eqn></inline-eqn> XXZ chains, such as Not, Hadamard, Pi-8, Phase and C-Not, with fidelity levels exceeding 99%.</p></abstract></abstract-group></header><body refstyle="numeric"><sec-level1 id="nj317881s1" label="1"><heading>Introduction</heading><p indent="no">For quantum computers to become a reality we need to find or build physical systems that faithfully implement the quantum gates used in the algorithms of quantum computation. The basic requirement is that the experimenter has access to two states of a quantum system that can be effectively decoupled from environmental noise for a sufficiently long time, and that transitions between these two states can be controlled to simulate a number of elementary quantum gates (unitary transformations). Systems that have been investigated intensively are atomic levels in ion traps [<cite linkend="nj317881bib1">1</cite>, <cite linkend="nj317881bib2">2</cite>], superconducting device physics using Josephson rings [<cite linkend="nj317881bib3">3</cite>], nuclear spins [<cite linkend="nj317881bib4">4</cite>] (using NMR in suitable molecules) and quantum dots [<cite linkend="nj317881bib5">5</cite>]. In this paper, we demonstrate the implementation of quantum gates using one-dimensional spin-<inline-eqn><math-text><italic>J</italic></math-text></inline-eqn> systems. The results are obtained using a computer simulation of these systems.</p><p>The Hamiltonian of the XXZ model with kink boundary conditions is given by
<display-eqn id="nj317881eqn1" textype="equation" notation="LaTeX" eqnnum="1" lines="multiline"></display-eqn>where <inline-eqn><math-text><italic>S</italic><sub>α</sub><sup>1</sup></math-text></inline-eqn>, <inline-eqn><math-text><italic>S</italic><sub>α</sub><sup>2</sup></math-text></inline-eqn> and <inline-eqn><math-text><italic>S</italic><sub>α</sub><sup>3</sup></math-text></inline-eqn> are the spin-<inline-eqn><math-text><italic>J</italic></math-text></inline-eqn> matrices acting on the site <inline-eqn><math-text>α</math-text></inline-eqn>. Apart from the magnitude of the spins, <inline-eqn><math-text><italic>J</italic></math-text></inline-eqn>, the main parameter of the model is the anisotropy <inline-eqn><math-text>Δ>1</math-text></inline-eqn> and the limit <inline-eqn><math-text>Δ→∞</math-text></inline-eqn> is referred to as the <italic>Ising limit</italic>. In the case of <inline-eqn></inline-eqn> kink boundary conditions were first introduced in [<cite linkend="nj317881bib6">6</cite>]. They lead to ground states with a domain wall between down spins on the left portion of the chain and up spins on the right. The third component of the magnetization, <inline-eqn><math-text><italic>S</italic><sup>3</sup><sub><upright>tot</upright></sub></math-text></inline-eqn>, is conserved, and there is exactly one ground state for each value of <inline-eqn><math-text><italic>M</italic></math-text></inline-eqn>. Different values of <inline-eqn><math-text><italic>M</italic></math-text></inline-eqn> correspond to different positions of the domain walls, which in one dimension are sometimes referred to as kinks. In [<cite linkend="nj317881bib7">7</cite>], Koma <italic>et al</italic> showed that there is a spectral gap above each of the ground states in this model for all values of <inline-eqn><math-text><italic>J</italic></math-text></inline-eqn>. Recently [<cite linkend="nj317881bib8">8</cite>] it was shown that for spin values <inline-eqn></inline-eqn> and for sufficiently large value of the anisotropy <inline-eqn><math-text>Δ</math-text></inline-eqn> the low lying spectrum of (<eqnref linkend="nj317881eqn1">1</eqnref>) for each value of <inline-eqn><math-text><italic>M</italic></math-text></inline-eqn> has isolated eigenvalues that persist in the thermodynamic limit.</p><p>The presence of isolated eigenvalues is ideal from the point of view of quantum computation. The idea is to use the subspace, denoted by <inline-eqn></inline-eqn>, of the ground state and the first excited state of the Hamiltonian to encode a qubit. In the absence of noise (coupling to the environment), states corresponding to eigenvalues have an infinite life time. Generically, when the eigenvalues are embedded in a continuum, arbitrarily small perturbations will turn them into resonances, i.e. states with a finite life time. By using a subspace of states corresponding to isolated eigenvalues, we can expect much larger life times even in the presence of noise. Heuristically, there is an energy barrier protecting the states from decaying. Since the eigenvalues are <italic>not</italic> protected by a topological invariant, it is possible to use local, finite-strength perturbations to control transitions in the system. In principle, such perturbations may be implemented in a suitable solid state setup.</p><p>Concretely the idea is to let the system evolve under its own unitary time evolution generated by the Hamiltonian (<eqnref linkend="nj317881eqn1">1</eqnref>) with the addition of a few local control fields. We have two requirements to fulfill: the time evolution should leave the qubit space <inline-eqn></inline-eqn> approximately invariant, and the (approximately) unitary matrix describing the dynamics restricted to <inline-eqn></inline-eqn> and stopped at a suitable time should coincide with the desired quantum gate.</p><p>The control inputs needed to drive the system such that high fidelity gates are obtained are determined using techniques from optimal control theory. The simulation of the time evolution of the chain that is large enough to resemble the properties in the thermodynamic limit is carried out using the density matrix renormalization group (DMRG) algorithm. Figure <figref linkend="nj317881fig1">1</figref> shows the transition of the magnetic profiles in the <inline-eqn><math-text><italic>z</italic></math-text></inline-eqn>-direction from the ground to the first excited state using the Not gate constructed from a spin-<inline-eqn></inline-eqn> XXZ spin chain of length 50 sites. We also demonstrate the construction of Pi-8, Hadamard and Phase gates that form a set of universal single qubit gates.</p><figure id="nj317881fig1" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="21.9pc" printcolour="no" filename="images/nj317881fig1.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj317881fig1.jpg"></graphic-file></graphic><caption type="figure" id="nj317881fc1" label="Figure 1"><p indent="no">The transitioning of the magnetization profile from the ground to the first excited state using a Not gate. Simulation obtained for a chain of 50 (<inline-eqn><math-text><italic>L</italic>=25</math-text></inline-eqn>) sites using DMRG with <inline-eqn><math-text>Δ<sup>−1</sup>=0.3</math-text></inline-eqn> and <inline-eqn><math-text><italic>M</italic>=0</math-text></inline-eqn>. The lines in between the ground and excited state profiles represent the profile at intermediate times between <inline-eqn><math-text><italic>t</italic>=0</math-text></inline-eqn> and the gate time <inline-eqn><math-text><italic>T</italic>=20</math-text></inline-eqn>.</p></caption></figure><p>In order to have a viable quantum computing scheme one needs to implement at least one two-qubit gate. Here, we have implemented the C-Not gate which, in combination with the one-qubit gates, is known to be universal [<cite linkend="nj317881bib9">9</cite>].</p><p>Our scheme capitalizes on the kink nature of the excitations of the XXZ Hamiltonian, which are rather sharply localized. We imagine a setup with two parallel chains with the location of the kink lined up in their ground states. The subspace for the two-qubit state space is then <inline-eqn></inline-eqn>, where <inline-eqn></inline-eqn> represents the space of isolated eigenvalues of the first chain and <inline-eqn></inline-eqn> for the second chain. A set of three controls localized near the kinks is used to generate the single qubit gates acting on <inline-eqn></inline-eqn> and <inline-eqn></inline-eqn> and a C-Not gate on <inline-eqn></inline-eqn>. This scheme produces a universal set of gates necessary for two-qubit computation. It is clear how to generalize this scheme to implement <inline-eqn><math-text><italic>n</italic></math-text></inline-eqn>-qubit computation. Since a universal set of single qubit gates and nearest neighbor C-Not gates are universal for <inline-eqn><math-text><italic>n</italic></math-text></inline-eqn>-qubit computation, this can be achieved by using <inline-eqn><math-text><italic>n</italic></math-text></inline-eqn> parallel chains and controls that are localized and act on neighboring chains only.</p><p>In the next section, we describe the model and review some of the past results. Then, in section <secref linkend="nj317881s3">3</secref>, the optimal control problem to construct the quantum gates is described. Section <secref linkend="nj317881s4">4</secref> is devoted to the DMRG algorithm and the specific adaptations to the XXZ spin chain. Finally, in section <secref linkend="nj317881s5">5</secref> we present our results based on numerical simulations of the XXZ Hamiltonian using the DMRG algorithm.</p></sec-level1><sec-level1 id="nj317881s2" label="2"><heading>The model</heading><p indent="no">In this section we describe in detail the spin-<inline-eqn><math-text><italic>J</italic></math-text></inline-eqn> ferromagnetic XXZ model with kink boundary conditions on the one-dimensional lattice <inline-eqn></inline-eqn>. The local Hilbert space for a single site <inline-eqn><math-text>α</math-text></inline-eqn> is <inline-eqn></inline-eqn> with <inline-eqn></inline-eqn>. We consider the Hilbert space for a finite chain on the sites <inline-eqn><math-text>[−<italic>L</italic>+1,<italic>L</italic>]={−<italic>L</italic>+1, −<italic>L</italic>+2,…, +<italic>L</italic>}</math-text></inline-eqn>. This is <inline-eqn></inline-eqn>. The Hamiltonian of the spin-<inline-eqn><math-text><italic>J</italic></math-text></inline-eqn> XXZ model is given by equation (<eqnref linkend="nj317881eqn1">1</eqnref>). Note that, by a telescoping sum, we can absorb the boundary fields into the local interactions:
<display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn>The main parameter of the model is the anisotropy <inline-eqn><math-text>Δ>1</math-text></inline-eqn> and we get the Ising limit as <inline-eqn><math-text>Δ→∞</math-text></inline-eqn>. It is mathematically more convenient to work with the parameter <inline-eqn><math-text>Δ<sup>−1</sup></math-text></inline-eqn>, which we then assume is in the interval <inline-eqn><math-text>[0, 1]</math-text></inline-eqn>. As we said, <inline-eqn><math-text>Δ<sup>−1</sup>=0</math-text></inline-eqn> is the Ising limit, and <inline-eqn><math-text>Δ<sup>−1</sup>=1</math-text></inline-eqn> is the isotropic XXX Heisenberg model. The Hamiltonian commutes with the total magnetization
<display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn>As indicated in the introduction, for each <inline-eqn><math-text><italic>M</italic>∈{−2<italic>JL</italic>, −2<italic>JL</italic>+1,…, 2<italic>JL</italic>}</math-text></inline-eqn>, the corresponding sector is defined to be the eigenspace of <inline-eqn><math-text><italic>S</italic><sup>3</sup><sub><upright>tot</upright></sub></math-text></inline-eqn> with eigenvalue <inline-eqn><math-text><italic>M</italic></math-text></inline-eqn>; clearly, these are invariant subspaces for all the Hamiltonians introduced above. These subspaces are called ‘sectors’.</p><p>It was shown [<cite linkend="nj317881bib6">6</cite>, <cite linkend="nj317881bib10">10</cite>, <cite linkend="nj317881bib11">11</cite>, <cite linkend="nj317881bib13">13</cite>, <cite linkend="nj317881bib14">14</cite>] that the kink boundary conditions lead to a family of ground states. It was also shown in [<cite linkend="nj317881bib10">10</cite>]–[<cite linkend="nj317881bib13">13</cite>] that for each sector there is a unique ground state of <inline-eqn><math-text><italic>H</italic><sub><italic>L</italic></sub><sup><upright>k</upright></sup>(Δ<sup>−1</sup>)</math-text></inline-eqn> with eigenvalue 0. Moreover, this ground state, <inline-eqn><math-text>ψ<sub><italic>M</italic></sub></math-text></inline-eqn>, is given by the following expression:
<display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn>where the sum is over all configurations for which <inline-eqn></inline-eqn> and the relationship between <inline-eqn><math-text>Δ>1</math-text></inline-eqn> and <inline-eqn><math-text><italic>q</italic>∈(0, 1)</math-text></inline-eqn> is given by <inline-eqn><math-text>Δ=(<italic>q</italic>+<italic>q</italic><sup>−1</sup>)/2</math-text></inline-eqn>. A straightforward calculation shows a sharp transition in the magnetization from fully polarized down at the left to fully polarized up at the right. For this reason they are called kink ground states. In [<cite linkend="nj317881bib7">7</cite>], Koma <italic>et al</italic> showed that there is a spectral gap above each of the ground states in this model for all values of <inline-eqn><math-text><italic>J</italic></math-text></inline-eqn>. Recently [<cite linkend="nj317881bib8">8</cite>], we were able to prove the following theorem.</p><proclaim id="nj317881pro1" type="theorem" format="num" style="italic"><heading>Theorem 1</heading><p indent="no">For spin values <inline-eqn></inline-eqn>, there exists a finite <inline-eqn><math-text>Δ<sub>0</sub></math-text></inline-eqn> so that for all <inline-eqn><math-text>Δ>Δ<sub>0</sub></math-text></inline-eqn>, the first excited state of the restriction of <inline-eqn><math-text><italic>H</italic><sub><italic>L</italic></sub><sup><upright>k</upright></sup>(Δ<sup>−1</sup>)</math-text></inline-eqn> to any sector with fixed magnetization, are isolated eigenvalues that persist in the thermodynamic limit. This first excited state is a simple eigenvalue except when <inline-eqn><math-text><italic>J</italic></math-text></inline-eqn> is an integer <inline-eqn><math-text>>1</math-text></inline-eqn> and <inline-eqn><math-text><italic>M</italic>=0 <upright>mod</upright> 2<italic>J</italic></math-text></inline-eqn>, in which case it is two-fold degenerate. As <inline-eqn><math-text><italic>J</italic></math-text></inline-eqn> increases, further isolated eigenvalues appear in the spectrum, which are also at most two-fold degenerate.</p></proclaim><p>In this paper, it was also proved that in certain values of spin and sector, for example <inline-eqn></inline-eqn> and <inline-eqn><math-text><italic>M</italic>=0</math-text></inline-eqn> both the ground and excited states are non-degenerate (simple eigenvalues). This is the qubit space we work with and our quantum gates will be unitaries on this space.</p></sec-level1><sec-level1 id="nj317881s3" label="3"><heading>Quantum gates using quantum control</heading><p indent="no">The problem of constructing quantum gates can be formulated as a problem in quantum control theory [<cite linkend="nj317881bib15">15</cite>]. The goal is to steer the system using a small number of control parameters such that the unitary operator describing the quantum dynamics after a finite time <inline-eqn><math-text><italic>T</italic></math-text></inline-eqn>, has maximal overlap with a desired target unitary (the gate). From a control perspective these problems reduce to control of bilinear systems evolving on finite-dimensional Lie groups. This is an optimal control problem on a two-level system which has been studied widely with exact results known in some cases. For example, time optimal implementation of single and two-qubit quantum gates was studied [<cite linkend="nj317881bib16">16</cite>] when the Lie algebra <inline-eqn><math-text><upright>g</upright></math-text></inline-eqn> of <inline-eqn><math-text><italic>su</italic>(2)</math-text></inline-eqn> (<inline-eqn><math-text><italic>su</italic>(4)</math-text></inline-eqn>) can be decomposed as a Cartan pair <inline-eqn><math-text><italic>g</italic>=<italic>k</italic>⊕<italic>p</italic></math-text></inline-eqn> where <inline-eqn><math-text><italic>k</italic></math-text></inline-eqn> is the Lie subalgebra generated by the drift Hamiltonian and <italic>p</italic> is the Lie subalgebra generated by the control Hamiltonians. Finding the time optimal trajectories is reduced to finding geodesics on the coset space <inline-eqn><math-text><italic>G</italic>/<italic>K</italic></math-text></inline-eqn> (<inline-eqn><math-text><italic>G</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>K</italic></math-text></inline-eqn> being the Lie Groups corresponding to <inline-eqn><math-text><italic>g</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>k</italic></math-text></inline-eqn>). The problem of driving the evolution operator while minimizing an energy-type quadratic cost was studied in [<cite linkend="nj317881bib17">17</cite>]. In this case the optimal solutions can be expressed as elliptic functions. The time optimal problem of population transfer problem of a two-level quantum system and bounded controls was studied in [<cite linkend="nj317881bib18">18</cite>] and again explicit expressions for the optimal trajectories. In this paper, we follow a numerical gradient-based approach to optimal control [<cite linkend="nj317881bib19">19</cite>, <cite linkend="nj317881bib20">20</cite>].</p><sec-level2 id="nj317881s3.1" label="3.1"><heading>Single qubit gates</heading><p indent="no">We consider the problem of time evolution of the one-dimensional XXZ chain under external controls. The equation of motion for the unitary evolution of the XXZ chain isolated from the environment is given by Schrödinger's equation
<display-eqn id="nj317881eqn2" textype="equation" notation="LaTeX" eqnnum="2"></display-eqn>In control terminology <inline-eqn><math-text><italic>H</italic><sub><italic>L</italic></sub><sup><upright>k</upright></sup>(Δ<sup>−1</sup>)</math-text></inline-eqn> is the free or drift Hamiltonian and <inline-eqn><math-text><italic>H</italic><sup><upright>ext</upright></sup></math-text></inline-eqn> is the control Hamiltonian corresponding to the control field <inline-eqn><math-text><italic>v</italic>(<italic>t</italic>)</math-text></inline-eqn>. We require that <inline-eqn></inline-eqn> is an invariant subspace of <inline-eqn><math-text><italic>H</italic><sup><upright>ext</upright></sup></math-text></inline-eqn>, so that the time evolution of the system <eqnref linkend="nj317881eqn2">2</eqnref> given by the unitary <inline-eqn><math-text><italic>U</italic> (<italic>t</italic>)</math-text></inline-eqn> starting from an initial state in <inline-eqn></inline-eqn> will be constrained to <inline-eqn></inline-eqn> at all future times. The induced evolution on <inline-eqn></inline-eqn> at any specified final time <inline-eqn><math-text><italic>T</italic></math-text></inline-eqn> will be the quantum gate on the qubit space <inline-eqn></inline-eqn> and is given by the <inline-eqn><math-text>2×2</math-text></inline-eqn> matrix
<display-eqn id="nj317881eqn3" textype="equation" notation="LaTeX" eqnnum="3"></display-eqn>The control Hamiltonian we choose is the two-site operator <inline-eqn><math-text><italic>H</italic><sup><upright>ext</upright></sup>=<italic>S</italic><sub>0</sub><sup>3</sup><italic>S</italic><sub>1</sub><sup>3</sup></math-text></inline-eqn>. In practice for <inline-eqn><math-text><italic>S</italic><sub>0</sub><sup>3</sup><italic>S</italic><sub>1</sub><sup>3</sup></math-text></inline-eqn> there is a very small error probability for states to move out of <inline-eqn></inline-eqn> and the matrix <inline-eqn><math-text><italic>U</italic><sub><italic>xxz</italic></sub></math-text></inline-eqn> is not exactly unitary. The matrix elements <inline-eqn><math-text>⟨ψ<sub>0</sub>|<italic>H</italic><sup><upright>ext</upright></sup>|ψ<sub><italic>k</italic></sub>⟩</math-text></inline-eqn> and <inline-eqn><math-text>⟨ψ<sub>1</sub>|<italic>H</italic><sup><upright>ext</upright></sup>|ψ<sub><italic>k</italic></sub>⟩</math-text></inline-eqn> <inline-eqn><math-text><italic>k</italic>≠0, 1</math-text></inline-eqn> are proportional to the transition probabilities to move from states <inline-eqn><math-text>ψ<sub>0</sub></math-text></inline-eqn> and <inline-eqn><math-text>ψ<sub>1</sub></math-text></inline-eqn> to other eigenstates of <inline-eqn><math-text><italic>H</italic><sub><italic>L</italic></sub><sup><upright>k</upright></sup>(Δ<sup>−1</sup>)</math-text></inline-eqn>. We calculate the error probability to move out of the subspace <inline-eqn></inline-eqn> by the following estimates of these matrix elements
<display-eqn id="nj317881eqn4" textype="equation" notation="LaTeX" eqnnum="4"></display-eqn>for <inline-eqn><math-text><italic>i</italic>=0, 1</math-text></inline-eqn>. Figure <figref linkend="nj317881fig2">2</figref> shows that the probabilities of transitioning out of the subspace <inline-eqn></inline-eqn> are extremely small for <inline-eqn><math-text>Δ<sup>−1</sup>⩽0.3</math-text></inline-eqn>.</p><figure id="nj317881fig2" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="21.9pc" printcolour="no" filename="images/nj317881fig2.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="no" filename="images/nj317881fig2.jpg"></graphic-file></graphic><caption type="figure" id="nj317881fc2" label="Figure 2"><p indent="no">ErrProb, defined in (<eqnref linkend="nj317881eqn4">4</eqnref>), as a function of <inline-eqn><math-text>Δ<sup>−1</sup></math-text></inline-eqn>, calculated for the ground and first excited state <inline-eqn><math-text>ψ<sub>0</sub></math-text></inline-eqn> and <inline-eqn><math-text>ψ<sub>1</sub></math-text></inline-eqn>, with <inline-eqn><math-text><italic>H</italic><sup><upright>ext</upright></sup>=<italic>S</italic><sub>0</sub><sup>3</sup><italic>S</italic><sub>1</sub><sup>3</sup></math-text></inline-eqn>. This quantity provides a measure of the escape rate out of the qubit subspace.</p></caption></figure></sec-level2><sec-level2 id="nj317881s3.2" label="3.2"><heading>Implementing two-qubit gates</heading><p indent="no">The idea for implementing two-qubit gates is to use two copies of the XXZ chain. The Hilbert space for two-qubit quantum computation is <inline-eqn></inline-eqn>, where <inline-eqn></inline-eqn> and <inline-eqn></inline-eqn> are the subspaces spanned by the ground state and the first excited state of the first chain and the second chain, respectively. The Hamiltonian of an uncoupled two chain system is given by
<display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn>Here the notation <inline-eqn><math-text><italic>H</italic><sub><italic>L</italic></sub><sup><upright>k</upright></sup>(Δ<sup>−1</sup>)<sup>(1)</sup></math-text></inline-eqn> is to be interpreted as <inline-eqn></inline-eqn> and <inline-eqn><math-text><italic>H</italic><sub><italic>L</italic></sub><sup><upright>k</upright></sup>(Δ<sup>−1</sup>)<sup>(2)</sup></math-text></inline-eqn> is to be interpreted as <inline-eqn></inline-eqn>. The two-qubit space is spanned by the four vectors <inline-eqn><math-text>ψ<sub><italic>mn</italic></sub>: =ψ<sub><italic>m</italic></sub>⊗ψ<sub><italic>n</italic></sub></math-text></inline-eqn> for <inline-eqn><math-text><italic>m</italic>,<italic>n</italic>=0, 1</math-text></inline-eqn> which are eigenvectors of the above Hamiltonian. If we consider the control system
<display-eqn id="nj317881eqn5" textype="equation" notation="LaTeX" eqnnum="5"></display-eqn>with <inline-eqn></inline-eqn>, then by selectively turning on <inline-eqn><math-text><italic>v</italic><sub>1</sub>(<italic>t</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>v</italic><sub>2</sub>(<italic>t</italic>)</math-text></inline-eqn> for certain time periods, the above system is equivalent to the control system (<eqnref linkend="nj317881eqn2">2</eqnref>) on chains 1 and 2, respectively, during those time intervals. This can be used to generate single qubit gates on <inline-eqn></inline-eqn> and <inline-eqn></inline-eqn>. Moreover, by simultaneously using <inline-eqn><math-text><italic>v</italic><sub>1</sub>(<italic>t</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>v</italic><sub>2</sub>(<italic>t</italic>)</math-text></inline-eqn> the local gates, i.e. gates of the kind <inline-eqn><math-text><italic>X</italic><sub>1</sub>⊗<italic>Y</italic><sub>2</sub></math-text></inline-eqn> can be generated on <inline-eqn></inline-eqn>. To implement a two-qubit quantum computing scheme we need to also implement perfectly entangling gates, i.e. a gate that can take a product state to a maximally entangled state. It is known that single qubit gates and any perfectly entangling gate are universal for two-qubit quantum computing [<cite linkend="nj317881bib21">21</cite>]. Clearly such a gate cannot be implemented by the control scheme (<eqnref linkend="nj317881eqn5">5</eqnref>) alone. In this paper, we choose to implement the C-Not gate, which is an example of a perfectly entangling gate. For this purpose we make use of an additional control namely <inline-eqn><math-text>(<italic>S</italic><sub>0</sub><sup>3</sup><italic>S</italic><sub>1</sub><sup>3</sup>)<sup>(1)</sup>⊗(<italic>S</italic><sub>0</sub><sup>3</sup><italic>S</italic><sub>1</sub><sup>3</sup>)<sup>(2)</sup></math-text></inline-eqn>.</p><p>We demonstrate the C-Not gate to high precision by using following control system
<display-eqn id="nj317881eqn6" textype="equation" notation="LaTeX" eqnnum="6" lines="multiline"></display-eqn>with <inline-eqn></inline-eqn> by selectively turning on and off some or all of the control fields <inline-eqn><math-text><italic>v</italic><sub>1</sub>(<italic>t</italic>)</math-text></inline-eqn>, <inline-eqn><math-text><italic>v</italic><sub>2</sub>(<italic>t</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>v</italic><sub>3</sub>(<italic>t</italic>)</math-text></inline-eqn> for specified time periods. Figure <figref linkend="nj317881fig3">3</figref> shows a diagrammatic representation of the two-qubit scheme. The C-Not gate is then given by the <inline-eqn><math-text>4×4</math-text></inline-eqn> matrix with elements
<display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn></p><figure id="nj317881fig3" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="18.9pc" printcolour="no" filename="images/nj317881fig3.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj317881fig3.jpg"></graphic-file></graphic><caption type="figure" id="nj317881fc3" label="Figure 3"><p indent="no">Configuration of two XXZ chains showing the localized controls required to implement the C-Not gate.</p></caption></figure></sec-level2><sec-level2 id="nj317881s3.3" label="3.3"><heading>Optimal control</heading><p indent="no">We first solve the control problems (<eqnref linkend="nj317881eqn2">2</eqnref>) and (<eqnref linkend="nj317881eqn6">6</eqnref>) for the projected system on <inline-eqn></inline-eqn> for the single chain and <inline-eqn></inline-eqn> for two chain system.</p><p><display-eqn id="nj317881eqn7" textype="equation" notation="LaTeX" eqnnum="7"></display-eqn>For the projected system on <inline-eqn></inline-eqn> the <inline-eqn><math-text><italic>H</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>B</italic><sub><italic>k</italic></sub></math-text></inline-eqn>s are given by the <inline-eqn><math-text>2×2</math-text></inline-eqn> matrices
<display-eqn id="nj317881eqn8" textype="equation" notation="LaTeX" eqnnum="8" lines="multiline"></display-eqn>whereas the projected system on <inline-eqn></inline-eqn> the control problem involves <inline-eqn><math-text>4×4</math-text></inline-eqn> matrices
<display-eqn id="nj317881eqn9" textype="equation" notation="LaTeX" eqnnum="9" lines="multiline"></display-eqn>where <inline-eqn><math-text><italic>m</italic>,<italic>n</italic>,<italic>r</italic>,<italic>s</italic>=0, 1</math-text></inline-eqn>. The overlap between a desired unitary gate <inline-eqn><math-text><italic>U</italic><sub><italic>f</italic></sub></math-text></inline-eqn> and the solution of (<eqnref linkend="nj317881eqn6">6</eqnref>) at time <inline-eqn><math-text><italic>T</italic></math-text></inline-eqn>, <inline-eqn><math-text><italic>U</italic>(<italic>T</italic>)</math-text></inline-eqn>, is measured as the difference in the norm square <inline-eqn><math-text>∥<italic>U</italic><sub><italic>f</italic></sub>−<italic>U</italic>(<italic>T</italic>)∥<sup>2</sup></math-text></inline-eqn>, and the norm is defined in terms of the standard inner product <inline-eqn><math-text>⟨<italic>V</italic>|<italic>W</italic>⟩: =<upright>T r</upright>(<italic>V</italic><sup>†</sup><italic>W</italic>)</math-text></inline-eqn>. The norm can be written as
<display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn>and hence minimizing this norm is equivalent to maximizing
<display-eqn id="nj317881eqn10" textype="equation" notation="LaTeX" eqnnum="10"></display-eqn>We define the gate fidelity as
<display-eqn id="nj317881eqn11" textype="equation" notation="LaTeX" eqnnum="11"></display-eqn>To select the optimal control fields <inline-eqn><math-text><italic>v</italic><sub><italic>i</italic></sub>(<italic>t</italic>)</math-text></inline-eqn> we use the numerical gradient ascent approach described in many books on control theory. This approach was applied to the quantum setting in [<cite linkend="nj317881bib20">20</cite>]. We start with the necessary conditions for optimality called the Pontryagin maximum principle which is a generalization of the Euler–Lagrange equations from calculus of variations. In the problems with costs of type (<eqnref linkend="nj317881eqn10">10</eqnref>) and no <italic>a priori</italic> bound on controls, Pontryagin's maximum principle takes the following form.</p><proclaim id="nj317881pro2" type="theorem" format="num" style="italic"><heading>Theorem 2</heading><p indent="no">(Pontryagin maximum principle [<cite linkend="nj317881bib20">20</cite>, <cite linkend="nj317881bib22">22</cite>]). If <inline-eqn><math-text><italic>v</italic><sub><italic>i</italic></sub>(<italic>t</italic>)<italic>s</italic></math-text></inline-eqn> are optimal controls of the system (<eqnref linkend="nj317881eqn6">6</eqnref>) and <inline-eqn><math-text><italic>U</italic> (<italic>t</italic>)</math-text></inline-eqn> the corresponding trajectory solution, then there exists a nonzero operator valued Lagrange multiplier <inline-eqn><math-text>λ</math-text></inline-eqn> which is the solution of the adjoint equations
<display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn>and a scalar valued Hamiltonian function <inline-eqn><math-text><italic>h</italic>(<italic>U</italic> (<italic>t</italic>), <italic>v</italic><sub><italic>i</italic></sub>(<italic>t</italic>)): =<upright>Re</upright> <upright>T r</upright>(−<upright>i</upright>λ<sup>′</sup>(<italic><italic><italic>t</italic></italic></italic>)<italic>H</italic> (<italic><italic><italic>t</italic></italic></italic>)<italic>U</italic> (<italic><italic><italic>t</italic></italic></italic>))</math-text></inline-eqn> such that, for every <inline-eqn><math-text>τ∈(0, <italic>T</italic>]</math-text></inline-eqn> we have
<display-eqn id="nj317881eqn12" textype="equation" notation="LaTeX" eqnnum="12"></display-eqn></p></proclaim><p indent="no">The algorithm to find the optimal controls is as follows:
<ordered-list id="list1" type="roman" pattern="2"><list-item id="list1-i1" marker="(i)"><p>A suitable gate time <inline-eqn><math-text><italic>T</italic></math-text></inline-eqn> is chosen and discretized in <inline-eqn><math-text><italic>N</italic></math-text></inline-eqn> equal steps of duration <inline-eqn></inline-eqn>. The initial control <inline-eqn><math-text><italic>v</italic><sub><italic>i</italic></sub><sup>(0)</sup>(<italic>t</italic><sub><italic>k</italic></sub>)</math-text></inline-eqn> for all the discretized time intervals is based on a guess or at random.</p></list-item><list-item id="list1-i2" marker="(ii)"><p>For these piecewise constant controls, from <inline-eqn></inline-eqn> and <inline-eqn><math-text>λ(<italic>T</italic>)=−<italic>U</italic><sub><italic>f</italic></sub></math-text></inline-eqn>, compute the forward and backward propagation respectively as follows
<display-eqn id="nj317881eqn13" textype="equation" notation="LaTeX" eqnnum="13"></display-eqn><display-eqn id="nj317881eqn14" textype="equation" notation="LaTeX" eqnnum="14"></display-eqn>for all <inline-eqn><math-text><italic>t</italic><sub>1</sub>, …, <italic>t</italic><sub><italic>N</italic></sub></math-text></inline-eqn> and where <inline-eqn><math-text><italic>r</italic></math-text></inline-eqn> is an iteration number of the algorithm initially set to 0 and
<display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn></p></list-item><list-item id="list1-i3" marker="(iii)"><p>Substitute the equations (<eqnref linkend="nj317881eqn13">13</eqnref>) and (<eqnref linkend="nj317881eqn14">14</eqnref>) into equation (<eqnref linkend="nj317881eqn12">12</eqnref>) to evaluate the gradient, and then update the controls as
<display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn>where <inline-eqn><math-text>τ</math-text></inline-eqn> is a small step size.</p></list-item><list-item id="list1-i4" marker="(iv)"><p>if <inline-eqn></inline-eqn> (<inline-eqn><math-text>γ</math-text></inline-eqn> being the level of accuracy) then done, otherwise go to step (2) for the next iteration with the updated controls.</p></list-item></ordered-list>Having solved the control problem on the projected systems to get the optimal controls <inline-eqn><math-text><italic>v</italic><sub>1</sub>(<italic>t</italic>)</math-text></inline-eqn>, <inline-eqn><math-text><italic>v</italic><sub>2</sub>(<italic>t</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>v</italic><sub>3</sub>(<italic>t</italic>)</math-text></inline-eqn> we would like to apply them to a large system and see their effects on the projected system. However simulating even a moderately sized spin chain is hard because of the exponentially growing dimension of the Hilbert space. In the next section we describe an algorithm by which we are able to simulate the XXZ chain of 50 sites.</p></sec-level2></sec-level1><sec-level1 id="nj317881s4" label="4"><heading>DMRG simulations for quantum gates</heading><p indent="no">To see the effect of the evolution of the XXZ chain with external magnetic controls we numerically simulate the XXZ chain using the DMRG algorithm. The dynamics of the interfaces of the XXZ chain using DMRG was studied recently in [<cite linkend="nj317881bib23">23</cite>]. The standard DMRG algorithm is a numerical algorithm originally developed by White [<cite linkend="nj317881bib24">24</cite>] that has worked successfully in providing very accurate results for ground state energies and correlation functions in strongly correlated systems. Modifications to this method [<cite linkend="nj317881bib25">25</cite>, <cite linkend="nj317881bib26">26</cite>] allow to address the physics of time-dependent and out of equilibrium systems. The crux of the DMRG algorithm is a decimation procedure that chooses the physically most relevant states to describe the target states. It is now known that DMRG works well because the ground states of non-critical quantum chains like the XXZ chain are only slightly entangled, i.e. they obey an area law of entanglement that says that the entanglement between a distinguished block of the chain and the rest of the chain is bounded by the boundary area of the block. In fact the DMRG procedure is a variational ansatz over states known as Matrix product states (MPS) [<cite linkend="nj317881bib27">27</cite>]. The standard DMRG procedure and its connection with MPS and entanglement is described in detail in [<cite linkend="nj317881bib28">28</cite>]. For a single XXZ chain our target states are the ground state <inline-eqn><math-text>ψ<sub>0</sub></math-text></inline-eqn> and first excited state <inline-eqn><math-text>ψ<sub>1</sub></math-text></inline-eqn> restricted to a sector of magnetization. We use the standard DMRG procedure with the adaptation that we grow the chain while restricting the blocks to the sector of zero magnetization using the symmetry of the Hamiltonian (see [<cite linkend="nj317881bib23">23</cite>]).</p><p>For the two-qubit gates we convert the two chain system to a one dimensional spin chain by a spin ladder construction.</p><p><display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn><display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn>The single site Hilbert space for the DMRG is the rung composed of <inline-eqn></inline-eqn> for <inline-eqn><math-text>α∈[−<italic>L</italic>+1…<italic>L</italic>]</math-text></inline-eqn>. On this site we define the local operators
<display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn>We can then write the Hamiltonian of this single chain using the above construction
<display-eqn id="nj317881eqn15" textype="equation" notation="LaTeX" eqnnum="15" lines="multiline"></display-eqn><display-eqn id="nj317881eqn16" textype="equation" notation="LaTeX" eqnnum="16" lines="multiline"></display-eqn>for <inline-eqn><math-text><italic>k</italic>=1, 2</math-text></inline-eqn>. We carry out the DMRG procedure as described in the algorithm with the Hamiltonian <inline-eqn><math-text><italic>H</italic><sub><italic>L</italic></sub><sup><upright>k</upright></sup>(Δ<sup>−1</sup>)<sup>(1, 2)</sup></math-text></inline-eqn> but we ensure that we keep both the chains in the magnetization sector 0 by simultaneously diagonalizing <inline-eqn><math-text><italic>H</italic><sub><italic>L</italic></sub><sup><upright>k</upright></sup>(Δ<sup>−1</sup>)<sup>(1, 2)</sup></math-text></inline-eqn> with the total magnetization operators
<display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn>The target states <inline-eqn><math-text>ψ<sub>0</sub>⊗ψ<sub>0</sub></math-text></inline-eqn>, <inline-eqn><math-text>ψ<sub>0</sub>⊗ψ<sub>1</sub></math-text></inline-eqn>, <inline-eqn><math-text>ψ<sub>1</sub>⊗ψ<sub>0</sub></math-text></inline-eqn>, <inline-eqn><math-text>ψ<sub>1</sub>⊗ψ<sub>1</sub></math-text></inline-eqn> are the simultaneous eigenvectors of the these operators and form the computational basis <inline-eqn><math-text>|00⟩</math-text></inline-eqn>, <inline-eqn><math-text>|01⟩</math-text></inline-eqn>, <inline-eqn><math-text>|10⟩</math-text></inline-eqn> and <inline-eqn><math-text>|11⟩</math-text></inline-eqn> for two-qubit quantum computation.</p><p>To compute the time evolution of the chain under the controlled evolution by the external fields we use the time-dependent DMRG procedure. The idea is that a two-site operator can be applied to a DMRG state most effectively by expressing the state in the basis where the left block has length <inline-eqn><math-text><italic>x</italic>−1</math-text></inline-eqn> so the two middle sites that are untruncated are the sites where the operator is acting. We can write the time evolution in the Trotter decomposition
<display-eqn id="nj317881eqn17" textype="equation" notation="LaTeX" eqnnum="17" lines="multiline"></display-eqn>To apply <inline-eqn><math-text><upright>e</upright><sup>−<upright>i</upright><italic>H</italic>δ</sup></math-text></inline-eqn> to the ground and excited states in the basis with the center sites all the way to the left we apply <inline-eqn><math-text><upright>e</upright><sup>−(<upright>i</upright>/2)h<sub>−<italic>L</italic>+1, −<italic>L</italic>+2</sub></sup></math-text></inline-eqn>. After shifting one site to the right we apply <inline-eqn><math-text><upright>e</upright><sup>−(<upright>i</upright>/2)h<sub>−<italic>L</italic>+2, −<italic>L</italic>+3</sub></sup></math-text></inline-eqn> etc. Since all our controls are two site controls at the center, only the interaction <inline-eqn><math-text><italic>h</italic><sub>0, 1</sub></math-text></inline-eqn> is time-dependent. In the adaptive time-dependent methods the Hilbert space is continuously modified as time progresses by carrying out reduced basis transformations on the evolved state. In our case since the gates are obtained in a relatively short period of time our Hilbert space remains unchanged resembling the static DMRG methods.</p></sec-level1><sec-level1 id="nj317881s5" label="5"><heading>Results</heading><p indent="no">In this section, we present numerical results of the construction of quantum gates using the spin- <inline-eqn></inline-eqn> XXZ spin chain. Our results are for the universal set of single qubit gates consisting of the Not (X), Hadamard (H), Pi-8 (T) and Phase (S) gates and the two-qubit C-Not gate. All results are obtained using the DMRG algorithm and the optimal control methods described in the previous sections. The steps carried out to obtain the single qubit gates are as follows:
<ordered-list id="list2" type="roman" pattern="2"><list-item id="list2-i5" marker="(i)"><p>We use ground state DMRG of the XXZ chain to obtain the lowest eigenvectors <inline-eqn><math-text>ψ<sub>0</sub></math-text></inline-eqn> and <inline-eqn><math-text>ψ<sub>1</sub></math-text></inline-eqn> of <inline-eqn><math-text><italic>H</italic><sub><italic>L</italic></sub><sup><upright>k</upright></sup>(Δ<sup>−1</sup>)</math-text></inline-eqn> in the sector corresponding to <inline-eqn><math-text><italic>M</italic>=0</math-text></inline-eqn>.</p></list-item><list-item id="list2-i6" marker="(ii)"><p>We obtain the projected <inline-eqn><math-text>2×2</math-text></inline-eqn> control system of equation (<eqnref linkend="nj317881eqn7">7</eqnref>) with matrices <inline-eqn><math-text><italic>H</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>B</italic><sub>1</sub></math-text></inline-eqn> with matrix elements given by (<eqnref linkend="nj317881eqn8">8</eqnref>). For a target gate <inline-eqn><math-text><italic>U</italic><sub><italic>f</italic></sub></math-text></inline-eqn> and a suitable final time <inline-eqn><math-text><italic>T</italic></math-text></inline-eqn> we find the optimal control <inline-eqn><math-text><italic>v</italic><sub>1</sub>(<italic>t</italic>)</math-text></inline-eqn> on this <inline-eqn><math-text>2×2</math-text></inline-eqn> system using the technique described in section <secref linkend="nj317881s3.3">3.3</secref>.</p></list-item><list-item id="list2-i7" marker="(iii)"><p>Finally we apply the time-dependent DMRG procedure of section <secref linkend="nj317881s4">4</secref> to the chain of (<eqnref linkend="nj317881eqn2">2</eqnref>) for a specified time <inline-eqn><math-text><italic>T</italic></math-text></inline-eqn> starting from <inline-eqn><math-text>ψ<sub>0</sub></math-text></inline-eqn> and <inline-eqn><math-text>ψ<sub>1</sub></math-text></inline-eqn> and using the <inline-eqn><math-text><italic>v</italic><sub>1</sub>(<italic>t</italic>)</math-text></inline-eqn> found in step 2 to get the time evolved states <inline-eqn><math-text>ψ<sub>0</sub>(<italic>T</italic>)=<italic>U</italic>(<italic>T</italic>)ψ<sub>0</sub></math-text></inline-eqn> and <inline-eqn><math-text>ψ<sub>1</sub>(<italic>T</italic>)=<italic>U</italic>(<italic>T</italic>)ψ<sub>1</sub></math-text></inline-eqn>. We compute the induced evolution on the subspace <inline-eqn></inline-eqn> to obtain the gate <inline-eqn><math-text><italic>U</italic><sub><italic>xxz</italic></sub></math-text></inline-eqn> given by the matrix elements <inline-eqn><math-text>⟨ψ<sub><italic>i</italic></sub>|ψ<sub><italic>j</italic></sub>(<italic>T</italic>)⟩</math-text></inline-eqn> for <inline-eqn><math-text><italic>i</italic>,<italic>j</italic>=0, 1</math-text></inline-eqn> and compare the overlap with <inline-eqn><math-text><italic>U</italic><sub><italic>f</italic></sub></math-text></inline-eqn> using equation (<eqnref linkend="nj317881eqn11">11</eqnref>).</p></list-item></ordered-list>Our desired single-qubit target gates are given by the unitaries.</p><p><display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn>Note that the <inline-eqn><math-text><italic>X</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>H</italic></math-text></inline-eqn> gates have a extra factor <inline-eqn><math-text><italic>i</italic></math-text></inline-eqn> with respect to the conventional definition. This turned out to be convenient for us but is otherwise not important. The gates obtained using the XXZ chain and their fidelities are as follows. The optimal controls <inline-eqn><math-text><italic>v</italic><sub>1</sub>(<italic>t</italic>)</math-text></inline-eqn> used to get the gate results are shown in figure <figref linkend="nj317881fig4">4</figref> and table <tabref linkend="nj317881tab1">1</tabref>.
<display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn></p><figure id="nj317881fig4" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="31.9pc" printcolour="no" filename="images/nj317881fig4.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="no" filename="images/nj317881fig4.jpg"></graphic-file></graphic><caption type="figure" id="nj317881fc4" label="Figure 4"><p indent="no">(Clockwise) The controls for Not, Hadamard, Pi-8 and Phase gates plotted versus time discretized for time steps of <inline-eqn><math-text>Δ<italic>t</italic>=0.5</math-text></inline-eqn>.</p></caption></figure><table id="nj317881tab1" frame="topbot" position="float" width="fit" place="top"><caption type="table" id="nj317881tc1" label="Table 1"><p>Numerical simulation of the construction of the Not, Pi-8, Hadamard and Phase gates. Results are obtained using DMRG and time-dependent DMRG for a spin-<inline-eqn></inline-eqn> chain of <inline-eqn><math-text><italic>L</italic>=50</math-text></inline-eqn> sites and at <inline-eqn><math-text>Δ<sup>−1</sup>=0.3</math-text></inline-eqn> in the sector corresponding to <inline-eqn><math-text><italic>M</italic>=0</math-text></inline-eqn>. The table shows the values of the control field <inline-eqn><math-text><italic>v</italic><sub>1</sub>(<italic>t</italic>)</math-text></inline-eqn> with gate time <inline-eqn><math-text><italic>T</italic>=10</math-text></inline-eqn> discretized with <inline-eqn><math-text>Δ<italic>t</italic>=0.5</math-text></inline-eqn>.</p></caption><tgroup cols="4"><colspec colnum="1" colname="col1" align="left"></colspec><colspec colnum="2" colname="col2" align="left"></colspec><colspec colnum="3" colname="col3" align="left"></colspec><colspec colnum="4" colname="col4" align="left"></colspec><thead><row><entry>Not (X)</entry><entry>Hadamard (H)</entry><entry>Pi-8 (T)</entry><entry>Phase (S)</entry></row></thead><tbody><row><entry>0.1874</entry><entry><inline-eqn><math-text>−0.2182</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.1152</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.0797</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.0533</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.1176</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.2544</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.1889</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.2447</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.0631</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3310</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.2579</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.3587</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.0670</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3613</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.294</math-text></inline-eqn>5</entry></row><row><entry><inline-eqn><math-text>−0.3764</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.1296</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3632</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3085</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.2901</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.2396</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3524</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3091</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.1075</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3766</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3410</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3031</math-text></inline-eqn></entry></row><row><entry>0.1376</entry><entry><inline-eqn><math-text>−0.5154</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3358</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.2943</math-text></inline-eqn></entry></row><row><entry>0.3712</entry><entry><inline-eqn><math-text>−0.6286</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3383</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.2836</math-text></inline-eqn></entry></row><row><entry>0.4908</entry><entry><inline-eqn><math-text>−0.6917</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3443</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.2691</math-text></inline-eqn></entry></row><row><entry>0.4355</entry><entry><inline-eqn><math-text>−0.6899</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3441</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.2466</math-text></inline-eqn></entry></row><row><entry>0.2359</entry><entry><inline-eqn><math-text>−0.6241</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3222</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.2103</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.0246</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.5099</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.2590</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.1538</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.2681</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3723</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.1328</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.0715</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.4399</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.2404</math-text></inline-eqn></entry><entry>0.0709</entry><entry>0.0386</entry></row><row><entry><inline-eqn><math-text>−0.5065</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.1424</math-text></inline-eqn></entry><entry>0.3368</entry><entry>0.1704</entry></row><row><entry><inline-eqn><math-text>−0.4553</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.1001</math-text></inline-eqn></entry><entry>0.5877</entry><entry>0.3053</entry></row><row><entry><inline-eqn><math-text>−0.2959</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.1225</math-text></inline-eqn></entry><entry>0.7029</entry><entry>0.4128</entry></row><row><entry><inline-eqn><math-text>−0.0605</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.2033</math-text></inline-eqn></entry><entry>0.6294</entry><entry>0.4642</entry></row><row><entry>0.1909</entry><entry><inline-eqn><math-text>−0.3236</math-text></inline-eqn></entry><entry>0.4374</entry><entry>0.4516</entry></row></tbody></tgroup></table><p>For the C-Not gate the procedure described earlier is only slightly modified. We use the ground state DMRG of a one-dimensional chain built from the spin ladder described in section <secref linkend="nj317881s4">4</secref> to get four eigenvectors <inline-eqn><math-text>ψ<sub><italic>mn</italic></sub></math-text></inline-eqn> for <inline-eqn><math-text><italic>m</italic>,<italic>n</italic>=0, 1</math-text></inline-eqn>. The optimal control procedure is applied to the <inline-eqn><math-text>4×4</math-text></inline-eqn> control system (<eqnref linkend="nj317881eqn7">7</eqnref>) with <inline-eqn><math-text><italic>H</italic></math-text></inline-eqn>, <inline-eqn><math-text><italic>B</italic><sub>1</sub></math-text></inline-eqn>, <inline-eqn><math-text><italic>B</italic><sub>2</sub></math-text></inline-eqn>, <inline-eqn><math-text><italic>B</italic><sub>3</sub></math-text></inline-eqn> given by equations (<eqnref linkend="nj317881eqn9">9</eqnref>) to find the controls <inline-eqn><math-text><italic>v</italic><sub>1</sub>(<italic>t</italic>)</math-text></inline-eqn>, <inline-eqn><math-text><italic>v</italic><sub>2</sub>(<italic>t</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>v</italic><sub>3</sub>(<italic>t</italic>)</math-text></inline-eqn>. The time-dependent DMRG procedure is applied to the chain of equation (<eqnref linkend="nj317881eqn6">6</eqnref>) for time <inline-eqn><math-text><italic>T</italic></math-text></inline-eqn> with the controls <inline-eqn><math-text><italic>v</italic><sub>1</sub>(<italic>t</italic>)</math-text></inline-eqn>, <inline-eqn><math-text><italic>v</italic><sub>2</sub>(<italic>t</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>v</italic><sub>3</sub>(<italic>t</italic>)</math-text></inline-eqn> to get the time-evolved states <inline-eqn><math-text>ψ<sub><italic>mn</italic></sub>(<italic>T</italic>)=<italic>U</italic>(<italic>T</italic>)ψ<sub><italic>mn</italic></sub></math-text></inline-eqn>. The induced evolution on the subspace <inline-eqn></inline-eqn> gives the <inline-eqn><math-text><upright>C</upright>−<upright>Not</upright><sub><italic>xxz</italic></sub></math-text></inline-eqn> gate with matrix elements <inline-eqn><math-text>⟨ψ<sub><italic>mn</italic></sub>|ψ<sub><italic>rs</italic></sub>(<italic>T</italic>)⟩</math-text></inline-eqn>. Table <tabref linkend="nj317881tab2">2</tabref> shows the optimal controls <inline-eqn><math-text><italic>v</italic><sub>1</sub>(<italic>t</italic>)</math-text></inline-eqn>, <inline-eqn><math-text><italic>v</italic><sub>2</sub>(<italic>t</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>v</italic><sub>3</sub>(<italic>t</italic>)</math-text></inline-eqn> used to obtain the C-not gate. The gate obtained using the <inline-eqn><math-text><italic>XXZ</italic></math-text></inline-eqn> chain and gate fidelity is as follows:
<display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn><display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn><display-eqn textype="equation" notation="LaTeX" number="no"></display-eqn></p><table id="nj317881tab2" frame="topbot" position="float" width="fit" place="top"><caption type="table" id="nj317881tc2" label="Table 2"><p>The C-Not gate controls using two spin-<inline-eqn></inline-eqn> XXZ-chains of length <inline-eqn><math-text><italic>L</italic>=50</math-text></inline-eqn> at <inline-eqn><math-text>Δ<sup>−1</sup>=0.25</math-text></inline-eqn> and <inline-eqn><math-text><italic>M</italic>=0</math-text></inline-eqn> for both the chains. The gate time <inline-eqn><math-text><italic>T</italic>=3.5</math-text></inline-eqn> is discretized into <inline-eqn><math-text><italic>N</italic>=20</math-text></inline-eqn> time steps. The table shows the values for the three control fields <inline-eqn><math-text><italic>v</italic><sub>1</sub></math-text></inline-eqn>, <inline-eqn><math-text><italic>v</italic><sub>2</sub></math-text></inline-eqn> and <inline-eqn><math-text><italic>v</italic><sub>3</sub></math-text></inline-eqn> that are constant during any one of the time intervals.</p></caption><tgroup cols="3"><colspec colnum="1" colname="col1" align="left"></colspec><colspec colnum="2" colname="col2" align="left"></colspec><colspec colnum="3" colname="col3" align="left"></colspec><thead><row><entry><inline-eqn><math-text><italic>v</italic><sub>1</sub>(<italic>t</italic>)</math-text></inline-eqn></entry><entry><inline-eqn><math-text><italic>v</italic><sub>2</sub>(<italic>t</italic>)</math-text></inline-eqn></entry><entry><inline-eqn><math-text><italic>v</italic><sub>3</sub>(<italic>t</italic>)</math-text></inline-eqn></entry></row></thead><tbody><row><entry><inline-eqn><math-text>−0.4040</math-text></inline-eqn></entry><entry>0.3953</entry><entry>0.0540</entry></row><row><entry>2.4494</entry><entry>0.2588</entry><entry>0.0501</entry></row><row><entry>3.7163</entry><entry>0.1886</entry><entry><inline-eqn><math-text>−0.1314</math-text></inline-eqn></entry></row><row><entry>3.0455</entry><entry>0.1766</entry><entry><inline-eqn><math-text>−0.2677</math-text></inline-eqn></entry></row><row><entry>1.6565</entry><entry>0.2185</entry><entry><inline-eqn><math-text>−</math-text></inline-eqn>0.0872</entry></row><row><entry>0.5583</entry><entry>0.3085</entry><entry>0.1455</entry></row><row><entry><inline-eqn><math-text>−0.1036</math-text></inline-eqn></entry><entry>0.4206</entry><entry>0.2346</entry></row><row><entry><inline-eqn><math-text>−0.5117</math-text></inline-eqn></entry><entry>0.5196</entry><entry>0.1648</entry></row><row><entry><inline-eqn><math-text>−0.8215</math-text></inline-eqn></entry><entry>0.5716</entry><entry><inline-eqn><math-text>−0.0282</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−1.0083</math-text></inline-eqn></entry><entry>0.5578</entry><entry><inline-eqn><math-text>−0.3599</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.8630</math-text></inline-eqn></entry><entry>0.4926</entry><entry><inline-eqn><math-text>−0.9102</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.3774</math-text></inline-eqn></entry><entry>0.3612</entry><entry><inline-eqn><math-text>−1.4955</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.0810</math-text></inline-eqn></entry><entry>0.0978</entry><entry><inline-eqn><math-text>−1.4905</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.0940</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.1214</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.9196</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.0881</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.1138</math-text></inline-eqn></entry><entry><inline-eqn><math-text>−0.3217</math-text></inline-eqn></entry></row><row><entry><inline-eqn><math-text>−0.0408</math-text></inline-eqn></entry><entry>0.0794</entry><entry>0.0922</entry></row><row><entry><inline-eqn><math-text>−0.1805</math-text></inline-eqn></entry><entry>0.3399</entry><entry>0.3374</entry></row><row><entry><inline-eqn><math-text>−0.8094</math-text></inline-eqn></entry><entry>0.5868</entry><entry>0.4146</entry></row><row><entry><inline-eqn><math-text>−2.2183</math-text></inline-eqn></entry><entry>0.8522</entry><entry>0.2021</entry></row><row><entry><inline-eqn><math-text>−4.2425</math-text></inline-eqn></entry><entry>1.2694</entry><entry><inline-eqn><math-text>−0.4760</math-text></inline-eqn></entry></row></tbody></tgroup></table></sec-level1><acknowledgment><heading>Acknowledgments</heading><p indent="no">Based upon work supported in part by the National Science Foundation under grant nos DMS-0605342 and DMS-0757581. 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Phys.</jnl-title><volume>77</volume><pages>259</pages><crossref><cr_doi>http://dx.doi.org/10.1103/RevModPhys.77.259</cr_doi><cr_issn type="print">00346861</cr_issn><cr_issn type="electronic">15390756</cr_issn></crossref></journal-ref></reference-list></references></back></article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="eng"><title>Implementing quantum gates using the ferromagnetic spin-J XXZ chain with kink boundary conditions</title></titleInfo><titleInfo type="abbreviated"><title>Implementing quantum gates using the ferromagnetic spin-J XXZ chain</title></titleInfo><titleInfo type="alternative" lang="eng"><title>Implementing quantum gates using the ferromagnetic spin-J XXZ chain with kink boundary conditions</title></titleInfo><name type="personal"><namePart type="given">Tom</namePart><namePart type="family">Michoel</namePart><affiliation>Department of Plant Systems Biology, VIB, Department of Molecular Genetics, Ghent University, Technologiepark 927, B-9052 Gent, Belgium</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Jaideep</namePart><namePart type="family">Mulherkar</namePart><affiliation>Department of Mathematics, University of California, Davis, CA 95616-8633, USA</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Bruno</namePart><namePart type="family">Nachtergaele</namePart><affiliation>Department of Mathematics, University of California, Davis, CA 95616-8633, USA</affiliation><affiliation>Author to whom any correspondence should be addressed.</affiliation><affiliation>E-mail: bxn@math.ucdavis.edu</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="paper">paper</genre><originInfo><publisher>IOP Publishing</publisher><dateIssued encoding="w3cdtf">2010</dateIssued><copyrightDate encoding="w3cdtf">2010</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract>We demonstrate an implementation scheme for constructing quantum gates using unitary evolutions of the one-dimensional spin-J ferromagnetic XXZ chain. We present numerical results based on simulations of the chain using the time-dependent density matrix renormalization group method and techniques from optimal control theory. Using only a few control parameters, we find that it is possible to implement one- and two-qubit gates on a system of spin- XXZ chains, such as Not, Hadamard, Pi-8, Phase and C-Not, with fidelity levels exceeding 99.</abstract><relatedItem type="host"><titleInfo><title>New Journal of Physics</title></titleInfo><titleInfo type="abbreviated"><title>New J. Phys.</title></titleInfo><genre type="journal">journal</genre><identifier type="ISSN">1367-2630</identifier><identifier type="eISSN">1367-2630</identifier><identifier type="PublisherID">nj</identifier><identifier type="CODEN">NJOPFM</identifier><identifier type="URL">stacks.iop.org/NJP</identifier><part><date>2010</date><detail type="volume"><caption>vol.</caption><number>12</number></detail><detail type="issue"><caption>no.</caption><number>2</number></detail><extent unit="pages"><start>1</start><end>15</end><total>15</total></extent></part></relatedItem><identifier type="istex">443D39C954A16B7BB8BD220B153760B4AC0C0A86</identifier><identifier type="DOI">10.1088/1367-2630/12/2/025003</identifier><identifier type="articleID">317881</identifier><identifier type="articleNumber">025003</identifier><accessCondition type="use and reproduction" contentType="copyright">IOP Publishing and Deutsche Physikalische Gesellschaft</accessCondition><recordInfo><recordContentSource>IOP</recordContentSource><recordOrigin>IOP Publishing and Deutsche Physikalische Gesellschaft</recordOrigin></recordInfo></mods></metadata><enrichments><json:item><type>multicat</type><uri>https://api.istex.fr/document/443D39C954A16B7BB8BD220B153760B4AC0C0A86/enrichments/multicat</uri></json:item></enrichments><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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